A "bang-bang" controller is defined in optimal control theory as one operating in accordance with the control law under saturation constraints in which every control variable takes its maximum or minimum allowed value according to whether the sign of its coefficient is positive or negative. For example, the state-space representation of a single-input, single-output system is EQU x'(t)=Ax+bf(t) EQU y(t)=c(tr)x
where
x'(t)=dx/dt vector, PA1 c(tr)=transpose of c-vector, PA1 A is a matrix, and PA1 f(t) is the input value.
In a bang-bang control system, the value of f(t) would take on its maximum or minimum values according to its sign.
An example of such a system is the common thermostatically-controlled heating system found in most homes. The input values would be represented by 1 or 0, depending on whether the temperature were lower or higher, respectively, than the value set on the thermostat. When the temperature drops below the value set on the thermostat, the input value f(t) is 1, represented by a switch closure, which turns on the furnace. When the temperature rises above the value set on the thermostat, the switch opens representing f(t) as 0 and causing the furnace to shut off. In actual practice, a small magnet is used to hold the thermostat's bi-metal switch closed until the temperature rises several degrees above the temperature setting. This prevents the system from excessive "hunting," i.e., short-period cycling around the set value.
The thermostatically-controlled furnace is also an example of a feedback control system having a first-order lag term. This term represents the time delay between the turning on of the thermostat switch and the rise in temperature resulting from the consequence of turning on the furnace. First, there is a time delay for the heat source to heat the plenum in the furnace to a sufficiently high temperature before turning on the circulating fan. Next, there is another delay for the heated air to be circulated in the space being heated sufficiently to raise its temperature.
Although one of the easiest systems to understand, the bang-bang control system is the most difficult to analyze. Nonlinear design techniques include modeling or graphic procedures, or both. A bang-bang controller is usually presented graphically by trajectories that are caused, by manipulation of the system parameters, to approach as fast as possible some optimal point, e.g., the origin of the complex plane.
An example of a bang-bang servo system is disclosed in U.S. Pat. No. 3,924,170.
Linear control systems contain controllers that are classed as proportional, integral, or derivative, or any combination of the three. These are well known in the art as are the advantages and disadvantages of each for particular applications. An advantage of integral control systems is that they produce a signal that is proportional to the time integral of the input of the controller. In terms of transfer functions, it is equivalent to the addition of a zero on the negative real axis and a pole at the origin of the complex s-plane. It increases the order of the proportional system by one and reduces a constant steady-state error to zero, if the system is stable. On the other hand, it can make a higher-order system unstable if the point on the negative real axis of the complex s-plane of the added zero is not chosen correctly.
Proportional output controllers are also well known in the art. For example, U.S. Pat. No. 3,878,358 shows an output that is varied proportional to the instantaneous error in the desired output temperature. (It does not, however, measure the time duration of the temperature oscillation which is an important feature in the invention being disclosed.) Proportional output can be performed by varying the duty cycle of an output pulse train or by varying the number of total pulses (or cycles) in a fixed time frame.